How to attempt a Projectile Motion Question

Projectile motion is always tricky with students. I've been there I know, but really it's just a matter of dissecting up the motion of the "flying" object into straight across and up/down.

So let's say we're trying to help Peter on this one,

(I know I use this picture all the time but it just cracks me up!)

If we're looking down from above, we would observe the cannon ball to be moving in a straight line (horizontal motion).

We will also see that (neglecting air resistance) the cannonball moves in a straight line with constant velocity.
Hence by substituting your kinematics equations to this motion, we have the following:

a_x = 0

u_x = v_x

s_x = u_xt 

But if we're looking straight at the cannon (at a far enough distance of course) we would observe the cannon ball to go up and then come back down again (vertical motion). So this means that there is acceleration (since speed decreases to zero and the cannon ball actually makes a U-turn back down) and this acceleration is the acceleration of free fall g or 9.81 ms^-2.

Again, by substituting your kinematics equations to this motion, we have the following:

a_y = 9.81 ms^-2

v_y = u_y+ a_yt

v²_y = u²_y+ 2a_ys_y

s_y = u_yt + 1/2 a_yt² 

*** make sure that you define a direction and assign the correct signs to the different vectors!

So now we have 2 different sets of equations (horizontal & vertical) describing what is essentially the same motion (the same cannon ball moving in the same path).

This means that if we combine the two sets, we will thus be able to describe what is happening to the cannon ball at any time t.

Eg, the initial velocity of the cannonball u as it leaves the cannon can be described as follows,

magnitude:  u² = u²_x + u²_y

at angle to the horizontal of: tan^-1(u_y/u_x)

Same for the velocity at any given time t.

 *** remember that the time taken t is the same for both motion descriptions because they are describing the same motion path!

In summary, stuff you need to think about when given a question:

1. Identify all the data given as u_x , u_y , v_x , v_y , s_x , s_y 
2. Take note whether it is released at an angle (means  u_x , u_y>0) or horizontally (u_y = 0)
3. Is your initial velocity pointing up or down? (if up is +ve then g will be -ve)
4. Is your final displacement below or above your starting point? (if up is +ve then s below starting point will be -ve)

Let me know if this helped!